Primal-dual path following method for nonlinear semi-infinite programs with semi-definite constraints

TL;DR

This paper introduces two primal-dual path-following algorithms for solving nonlinear semi-infinite semi-definite programs with infinitely many convex constraints, emphasizing convergence properties and computational efficiency.

Contribution

The paper proposes novel algorithms that avoid solving SDPs directly, providing convergence analysis and integrating a SQP method with advanced scaling techniques for faster local convergence.

Findings

Weak* convergence to KKT points demonstrated

Superlinear local convergence achieved with the second method

Numerical experiments show improved efficiency over discretization methods

Abstract

In this paper, we propose two algorithms for nonlinear semi-infinite semi-definite programs with infinitely many convex inequality constraints, called SISDP for short. A straightforward approach to the SISDP is to use classical methods for semi-infinite programs such as discretization and exchange methods and solve a sequence of (nonlinear) semi-definite programs (SDPs). However, it is often too demanding to find exact solutions of SDPs. Our first approach does not rely on solving SDPs but on approximately following {a path leading to a solution, which is formed on the intersection of the semi-infinite region and the interior of the semi-definite region. We show weak* convergence of this method to a Karush-Kuhn-Tucker point of the SISDP under some mild assumptions and further provide with sufficient conditions for strong convergence. Moreover, as the second method, to achieve fast local convergence, we integrate a two-step sequential quadratic programming method equipped with Monteiro-Zhang scaling technique into the first method. We particularly prove two-step superlinear convergence of the second method using Alizadeh-Hareberly-Overton-like, Nesterov-Todd, and Helmberg-Rendle-Vanderbei-Wolkowicz/Kojima-Shindoh-Hara/Monteiro scaling directions. Finally, we conduct some numerical experiments to demonstrate the efficiency of the proposed method through comparison with a discretization method that solves SDPs obtained by finite relaxation of the SISDP.